Partial DIfferential Equations problems

this is another of my problemsShow that if C is a piecewise continuously differentiable closed curve bounding D then the problem
[tex] \nabla^2 u= -F(x,y) \ in\ D[/tex]
[tex] u = f \ on \ C_{1} [/tex]
[tex] \frac{\partial u}{\partial n} + \alpha u = 0 \ on \ C_{2} [/tex]where C1 is a part of C and C2 the remainder and where alpha is a positive constant, has at most one solution.

Perhaps i dont remember a theorem i should have learnt in ap revious class... or i am not familiar with it but what would i use the divergence theorem here?
i eman i can get it down to this
[tex] e^x \frac{\partial}{\partial x} (u + \frac{\partial u}{\partial x}) + e^y \frac{\partial}{\partial y} (u + \frac{\partial u}{\partial y}) = 0 [/tex]
but hereafter i am stuck, please do advise!

For the first problem, you might begin by assuming that two solutions exist which satisfy the differential equation and boundary conditions. The difference of the two solutions satisfies a simpler set of equations, right? Maybe this is a good place to start.

For the second problem, the original equation already looks like the divergence of a vector field in 2d. Maybe you should start from this observation.